Can migration from MS Access to SQL Server or ... make database faster? A company asked me to migrate their database from MS Access to SQL Server or MySQL because they think that on those engines their database will run faster. But I wonder if their problem isn't because of their database design rather than MS Access. 

I think that their database is smaller than 1GB large and it will remain always smaller than 10GB. Let me be pessimist 100GB! I'm afraid that the migration doesn't accelerate their query execution time and they would remain unsatisfied. I'd like to know if their slow interaction could be because of Access or I should  contact consult them to continue working with access and just change the designs, optimize network and the programs?

In which case should I think migrations towards non relational databases like MongoDB?
We can't really answer whether you should consider other platforms such as mongodb - we know nothing about the application or its data and whether such a platform would be appropriate. 

I can tell you that SQL Server has a lot more features geared toward performance and tuning than Access will ever have. Access is pretty simplistic and, while it can function as a multi-user database, it is not a true RDBMS and really is meant for the desktop space. 

Whether those features in SQL Server will make your company's specific application any faster, again, shrug, we just don't know.