I'm loving Avaza at the moment. I researched pretty heavily when I was looking for a tool to manage projects, track time, send estimates, send invoices and let clients pay me from an invoice with Stripe/PayPal. I tried Duet, Pancake and Solo, and they all lacked one or more of these features.
Avaza is really intuitive and just works for me. I use it to send estimates which the client can approve/deny, if they approve I convert it to an invoice. I track my hours and expenses on their project, and can easily add them to the invoice. I add all tasks and phases to the project board (much like Trello), and log time on each task. I have it connected with Paypal and Stripe so my clients can pay however they prefer.
It's been invaluable to me, and really affordable too.
You never mentioned why a change is desired. I understand it may be old, that it is a custom solution, and that the UI leaves something to be desired.
It leads me to think, they want something new, somewhat mainstream, with a modern UI, but besides a clunky interface, you don't mention why the change is happening.
You are looking for project billing software. There are 2 ways that I would go about it
If your a smaller company, option 1 might be a better solution. If your a larger company option 2 might have more benefits.
Agree with the others here, don't skimp on tools for managing your team. Another possible free option with paid plans could be Avaza, I use it for my freelance work and cannot fault it. Really clean and powerful, and I don't pay anything, yet.