What are your best suggestions for unearthing cross-sell opportunities as an account manager? Are there specific questions to ask, or is it more about customer research?

It starts with a deep understanding of the account/customer's growth needs, industry or marketplace trends, challenges and gaps. This will give you base understanding of their current state. 

Next is a thorough needs analysis with your account which will then give you a very good understanding of the situation which will then better help you to position and recommend your (cross sell) solution better. 

Remember, no body likes to be sold to. But if they trust you enough and understand the value behind your offering, they will eventually interact and purchase.

For those who have already done the best proof cases with your accounts and have already earned their trust, cross-sell becomes easy for a the first trusted point of contact for your account. 

Many top performers rely on a "white space worksheet" to identify and prioritize potential sales targets within an account. They explore the buyers underlying needs and determine possible connections between their expanded portfolio of products and the buying organizations key initiatives. They also determine how satisfied and firmly entrenched the customer is with a competitive product.

One of the biggest reasons I see people become challenged with cross-selling is they want to tell their customer about their new product/service. Instead, we need to be asking good questions and uncovering where there may be an issue and where we can help them with the new product/service.

Of course, it's also important to recognize where the decision is being made too. Sometimes a different product/service will require new buying influences.

Discovery never ends, as an account manager your client came to the business with an initial problem to solve. Once one problem is solved, others will  surface as next in line. Some key questions to uncover those might include:

-Now that you are starting to see success with X, how has that impacted Y? (Ex: Now that you have started seeing better customer insights from this research, how has that impacted your marketing content?)

-With better results for Y, what other challenges have come to light with Z? (Ex: With better marketing content from those insights, what other challenges have come to light with qualified lead generation for your sales teams?)

-What are some of the ways you are trying to solve for that now?

-What challenges or obstacles do you foresee with those solutions?

-What are you hoping to see for results from that, and by when?

First, the Territory Analysis process shows the AE the white space in their book of business and from the sales leader's strategy/guidance it's clear what the strategic focus is (eg growth in market share in a part of the portfolio).  AEs create  'target lists' as a result of their analysis. 
Second,  a Quarterly Business Review or Executive Review or a routine like that with clients, where the vendor shares updates on solution usage, trends, regulatory updates and plans for solutions and the same is asked of the client. 
Third, marketing team organizes industry roundtables, webinars, etc to existing clients. Sellers and marketing invite clients and where appropriate AEs get involved in the follow up. 
Fourth, encourage AE to create an outbound routine to build out the contacts at existing clients - one of the most fruitful things AEs ask here is 'who else should I be working with?' 
Hope this helps.