How can I increase the memory on the fly with the hardware management console on AIX 5 & 6?
You can only dynamically increase/decrease the memory for an LPAR within the partition profiles' configured minimum/maximum. ie: Minimum 1GB, Maximum 10GB.
Assuming you want to work within this range, and the system has available memory, increasing the allocation dynamically is simple. From the Server view, select the LPAR you want to reconfigure, and in the menu either next to the name or in the bottom pane, navigate to: Dynamic Logical Partitioning -> Memory -> Add or Remove.
From here edit the values for the profile, it will be pre-populated with the current values, and hit apply. Note, if the allocation or more specifically de-allocation is large, you may want to increase the timeout value, as this operation can take a long time. If your HMC doesn't have a timeout option, you're probably running on some older code. If it times out, just retry the operation, and eventually it will reach your value.
If you do not available free memory on the system, you can take memory from another lpar using the same process. You can use the Dynamic Logical Partitioning -> Memory -> Move option to take from one LPAR and place it on another.
You can confirm the changes realtime on the host using topas or nmon.